The Mahowald invariant is a highly nontrivial map (with indeterminacy) from the homotopy groups of spheres to itself with deep connections to chromatic homotopy theory. In this talk I will discuss a variant of the Mahowald invariant that can be computed using knowledge of the R-motivic stable homotopy groups of spheres, and discuss its comparison to the classical Mahowald invariant. This is joint work with Dan Isaksen.

Consider the following experiment: a deck with $m$ copies of $n$ different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ``feedback'' is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model).
\\
\\
In this talk we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in $n$, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation, which is roughly the number of cards one gets by naively guessing the same card type $mn$ times. This is joint work with Persi Diaconis, Ron Graham, and Xiaoyu He.

The direct summand conjecture, part of the homological conjectures in commutative algebra, was first proposed by Mel Hochster in 1973. The conjecture is easily proven when the given ring contains a field, but the ``mixed characteristic'' case seemed, for the most part, impenetrable. In 2016 Yves Andre announced a proof of the conjecture which utilizes the theory of perfectoid spaces. We will attempt to give a (somewhat) detailed proof of the DSC in a simplified scenario, and then give a general idea of the methods used to prove the theorem in the general case.

In this talk, we will discuss the existence theory of Kahler-Ricci flow when the Kahler metric has unbounded curvature. We will discuss some application of the Kahler-Ricci flow in the study of uniformization and the regularity of Gromov-Hausdorff limit. This is joint work with L.-F. Tam.

I will talk about some recent work determining the archimedean
components of certain Eisenstein series and CAP forms induced from the
long root parabolic of $G_2$. I will also discuss how these results are
being used in some work in progress on producing nonzero classes in
symmetric cube Selmer groups.

At the microscopic level, the dynamics of arbitrary networks of chemical reactions can be modeled as jump Markov processes whose sample paths converge, in the limit of large number of molecules, to the solutions of a set of algebraic ordinary differential equations. Fluctuations around these asymptotic trajectories and the corresponding phase transitions can in principle be studied through large deviations theory in path space, also called Wentzell-Freidlin (W-F) theory. However, the specific form of the jump rates for this family of processes does not satisfy the standard regularity assumptions imposed by such theory, and weaker conditions need to be developed to deal with the framework at hand. Using tools of Lyapunov stability theory we design sufficient conditions for the applicability of large deviations estimates to the asymptotics of the Markov process at hand. We then translate such conditions in terms of the topological structure of the chemical reaction network. This ultimately allows to define a large class of chemical reaction systems to which the estimates of interest can automatically be applied.

We introduce projective space over a field, and we investigate its
geometry. Along the way we find connections to many other fascinating
geometries including M\``{o}bius strips, the Hopf fibration, and the Fano
plane. We also discuss the basics of algebraic geometry in projective
space.

We define a localised Euler class for isotropic
sections, and isotropic cones, in SO(N) bundles. We use this to give an
algebraic definition of Borisov-Joyce sheaf counting invariants on
Calabi-Yau 4-folds. When a torus acts, we prove a localisation result.
This talk is based on the joint work with Richard. P. Thomas.